Scenario Reduction: Old and New Results

Transcription

1 Scenario Reduction: Old and New Results W. Römisch Humboldt-University Berlin Institute of Mathematics Berlin, Germany (H. Heitsch, R. Henrion, C. Küchler) Workshop in honor of Jitka Dupačová Charles University Prague, November 20, 2014

2 Introduction Most approaches for solving stochastic programs of the form { } min f 0 (x, ξ)p (dξ) : x X Ξ with a probability measure P on Ξ R d and a (normal) integrand f 0, require numerical integration techniques, i.e., replacing the integral by some quadrature formula n f 0 (x, ξ)p (dξ) p i f 0 (x, ξ i ), Ξ where p i = P ({ξ i }), n i=1 p i = 1 and ξ i Ξ, i = 1,..., n. Since f 0 is often expensive to compute, the number n should be as small as possible. i=1

3 With v(p ) and S(P ) denoting the optimal value and solution set of the stochastic program, respectively, the following estimates are known v(p ) v(q) sup f 0 (x, ξ)(p Q)(dξ) x X Ξ ) S(Q) S(P ) + Ψ P (sup f 0 (x, ξ)(p Q)(dξ), x X where X is assumed to be compact, Q is a probability distribution approximating P and the function Ψ P is the inverse of the growth function of the objective near the solution set, i.e., Ψ 1 P (t) := inf { Ξ Ξ } f 0 (x, ξ)p (dξ) v(p ) : x X, d(x, S(P )) t. Hence, the distance d F with F := {f 0 (x, ) : x X} d F (P, Q) := sup f(ξ)(p Q)(dξ) f F becomes important when approximating P. Ξ

4 For given n N and for the special case p i = 1 n, i = 1,..., n, the best possible choice of elements ξ i Ξ, i = 1,..., n (scenarios), is obtained by minimizing sup x X Ξ f 0 (x, ξ)p (dξ) n 1 n f 0 (x, ξ i ), i=1 i.e., by solving the best approximation problem where min d F(P, Q) Q P n (Ξ) P n (Ξ) := {Q : Q is a uniform probability measure with n scenarios}. It may be reformulated as a semi-infinite program. and is known as optimal quantization of P with respect to the function class F. Such optimal quantization problems of probability measures are often extremely difficult to solve. Idea: Enlarging the class F to the class of all Lipschitz functions with a uniform constant. But, then min Q P n (Ξ) d F(P, Q) = O(n 1 d ).

5 If the functions f 0 (x, ) belong to mixed Sobolev spaces, then the convergence rate min Q P n (Ξ) d F(P, Q) = O(n 1+δ ) (δ (0, 0.5]) can be achieved by certain randomized Quasi-Monte Carlo methods. The convergence rate can be improved if the functions f F satisfy a higher degree of smoothness. Aim of the talk: Solving the best approximation problem for discrete probability measures P having many scenarios and for function classes F, which are relevant for two-stage stochastic programs (scenario reduction). Additional motivation: Scenario reduction methods may be important for generating scenario trees form multistage stochastic programs.

6 Comparison of N = 256 Monte Carlo Mersenne Twister points and randomly binary shifted Sobol points in dimension d = 500, projection (8,9)

7 Linear two-stage stochastic programs min { } c, x + Φ(q(ξ), h(ξ) T (ξ)x)p (dξ) : x X, Ξ where c R m, Ξ and X are polyhedral subsets of R d and R m, respectively, P is a probability measure on Ξ and the s m-matrix T ( ), the vectors q( ) R m and h( ) R s are affine functions of ξ. Furthermore, Φ and D denote the infimum function of the linear second-stage program and its dual feasibility set, i.e., Φ(u, t) := inf{ u, y :W y = t, y Y } ((u, t) R m R s ) D := {u R m : {z R s : W z u Y } }, where q(ξ) R m are the recourse costs, W is the s m recourse matrix, W the transposed of W and Y the polar cone to the polyhedral cone Y.

8 Theorem: (Walkup-Wets 69) The function Φ(, ) is finite and continuous on the polyhedral set D W (Y ). Furthermore, the function Φ(u, ) is piecewise linear convex on the polyhedral set W (Y ) for fixed u D, and Φ(, t) is piecewise linear concave on D for fixed t W (Y ). Assumptions: (A1) relatively complete recourse: for any (ξ, x) Ξ X, h(ξ) T (ξ)x W (Y ); (A2) dual feasibility: q(ξ) D holds for all ξ Ξ. (A3) existence of second moments: Ξ ξ 2 P (dξ) < +. Note that (A1) is satisfied if W (Y ) = R s (complete recourse). In general, (A1) and (A2) impose a condition on the support of P. Extensions to certain random recourse models, i.e., to W (ξ), exist.

9 Idea: Extend the class F such that it covers all two-stage models. Fortet-Mourier metrics: (as canonical distances for two-stage models) ζ r (P, Q) := sup f(ξ)(p Q)(dξ) : f F r (Ξ), Ξ where r 1 (r {1, 2} if W (ξ) W ) F r (Ξ) := {f : Ξ R : f(ξ) f( ξ) c r (ξ, ξ), ξ, ξ Ξ}, c r (ξ, ξ) := max{1, ξ r 1, ξ r 1 } ξ ξ (ξ, ξ Ξ). Proposition: (Rachev-Rüschendorf 98) If Ξ is bounded, ζ r may be reformulated as transportation problem { } ζ r (P, Q) = inf ĉ r (ξ, ξ)η(dξ, d ξ):π 1 η =P, π 2 η =Q, Ξ Ξ where ĉ r is a metric (reduced cost) with ĉ r c r and given by { n 1 } ĉ r (ξ, ξ) := inf c r (ξ li, ξ li+1 ) : n N, ξ li Ξ, ξ l1 = ξ, ξ ln = ξ. i=1

10 Scenario reduction We consider discrete distributions P with scenarios ξ i and probabilities p i, i = 1,..., N, and Q being supported by a given subset of scenarios ξ j, j J {1,..., N}, of P. Optimal reduction of a given scenario set J: The best approximation of P with respect to ζ r by such a distribution Q exists and is denoted by Q. It has the distance D J := ζ r (P, Q ) = min ζ r(p, Q) = p i min ĉr(ξ i, ξ j ) Q j J i J and the probabilities qj = p j + p i, j J, where i J j J j := {i J : j = j(i)} and j(i) arg min ĉr(ξ i, ξ j ), i J j J (optimal redistribution).

11 Determining the optimal index set J with prescribed cardinality N n is, however, a combinatorial optimization problem: (n-median problem) min {D J : J {1,..., N}, J = N n} Hence, the problem of finding the optimal set J for deleting scenarios is N P- hard and polynomial time algorithms are not available. First idea: Reformulation as linear mixed-integer program N j=1,j i N min n 1 p j x ij ĉ r (ξ i, ξ j ) x ij + y i = 1 i,j=1 (i = 1,..., N), s.t. N y i = n, i=1 x ij y i 0 x ij 1 (i, j = 1,..., N), y i {0, 1} (1,..., N). and application of standard software or of specialized algorithms. Solution: x ij = { mini J ĉ r (ξ i,ξ j ) nĉ r (ξ i,ξ j ) 0, else., i J, j J y i = { 1, i J 0, i J.

12 Fast reduction heuristics Second idea: Application of (randomized) greedy heuristics. Starting point (n = N 1): min p l min ĉ r (ξ l, ξ j ) l {1,...,N} j l Algorithm 1: (Backward reduction) Step [0]: J [0] :=. Step [i]: Step [N-n+1]: l i arg min l J [i 1] k J [i 1] {l} J [i] := J [i 1] {l i }. Optimal redistribution. p k min ĉ r (ξ k, ξ j ). j J [i 1] {l}

13 Starting point (n = 1): min u {1,...,N} N p k ĉ r (ξ k, ξ u ) k=1 Algorithm 2: (Forward selection) Step [0]: Step [i]: Step [n+1]: J [0] := {1,..., N}. u i arg min u J [i 1] J [i] := J [i 1] \ {u i }. k J [i 1] \{u} Optimal redistribution. p k min ĉ r (ξ k, ξ j ), j J [i 1] \{u}

14 Application: Optimization of gas transport in a huge transportation network including hundreds of gas delivery nodes. A stationary situation is considered; more than 8 years of hourly data available at all delivery nodes; multivariate probability distribution for the gas output in certain temperature classes is estimated; 10 3 samples based on randomized Quasi-Monte Carlo methods are generated and later reduced by scenario reduction to 50 scenarios Hourly mean daily power in kwh/h Mean daily temperature in C

15 Mixed-integer two-stage stochastic programs We consider min where Φ is given by { } c, x + Φ(q(ξ), h(ξ) T (ξ)x)p (dξ) : x X, Ξ Φ(u, t) := inf { u 1, y 1 + u 2, y 2 W 1y 1 + W 2 y 2 t y 1 R m 1 +, y 2 Z m 2 + for all pairs (u, t) R m 1+m 2 R r, and c R m, X is a closed subset of R m, Ξ a polyhedron in R s, W 1 Q r m 1, W 2 Q r m 2, and T (ξ) R r m, q(ξ) R m 1+m 2 and h(ξ) R r are affine functions of ξ, and P is a probability measure. We again assume (A1) for W = (W 1, W 2 ) (relatively complete recourse), (A2) (dual feasibility) and (A3). }

16 Example: (Schultz-Stougie-van der Vlerk 98) Stochastic multi-knapsack problem: min = max, m = 2, m 1 = 0, m 2 = 4, c = (1.5, 4), X = [ 5, 5] 2, h(ξ) = ξ, q(ξ) q = (16, 19, 23, 28), y i {0, 1}, i = 1, 2, 3, 4, P U(5, 5.5,..., 14.5, 15} (discrete) Second stage problem: MILP with variables and 882 constraints. T = ( ) W = ( )

17 The function Φ is well understood and the function class F r,b (Ξ) := {f1l B : f F r (Ξ), B B}, is relevant, where r {1, 2}, B is a class of (convex) polyhedra in Ξ and 1l B denotes the characteristic function of the set B. The class B contains all polyhedra of the form B = {ξ Ξ : h(ξ) T (ξ)x D}, where x X and D is a polyhedron in R s each of whose facets, i.e., (s 1)- dimensional faces, is parallel to a facet of the cone W 1 (R m 1 + ) or of the unit cube [0, 1] s. Hence, B is very problem-specific. Therefore, we consider alternatively the class of rectangular sets B rect = {I 1 I 2 I d : I j is a closed interval in R} covering the situation of pure integer programs and serving as heuristic for the general case.

18 Proposition: In case F = F r,brect (Ξ), the metric d F allows the estimates d F (P, Q) max{α Brect (P, Q), ζ r (P, Q)} ( ) d F (P, Q) C ζ r (P, Q) + α Brect (P, Q) s+1 1 where C is some constant only depending on Ξ and α Brect discrepancy given by is the rectangular α Brect (P, Q) := sup P (B) Q(B) B B rect If the set Ξ is bounded, even the estimate holds α Brect (P, Q) d F (P, Q) Cα Brect (P, Q) 1 s+1. Since α Brect has a stronger influence on d F than ζ r, we consider the composite distance d λ (P, Q) = λ α Brect (P, Q) + (1 λ) ζ r (P, Q) with λ [0, 1] close to 1.

19 Scenario reduction We consider again discrete distributions P with scenarios ξ i and probabilities p i, i = 1,..., N, and Q being supported by a subset of scenarios ξ j, j J {1,..., N}, of P with weights q j, j J, where J has cardinality N n. The problem of optimal scenario reduction consists in determining such a probability measure Q deviating from P as little as possible with respect to d λ. It can be written as min {d λ ( P, j J q jδ ξj ) J {1,..., N}, J = N n q j 0 j J, j J q j = 1 }. This optimization problem may be decomposed into an outer problem for determining the index set J and an inner problem for choosing the probabilities q j, j J.

20 From 10 4 Monte Carlo samples of a two-dimensional standard normal distribution 20 scenarios are selected that represent best approximations with respect to the first order Fortet-Mourier metric (left) and the Kolmogorov distance (right), i.e., the uniform distance of probability distribution functions.

21 To this end, we denote d(p, (J, q)) := d λ ( P, j J q jδ ξj ) S n := {q R n : q j 0, j J, j J q j = 1}. Then the optimal scenario reduction problem may be rewritten as min{min d(p, (J, q)) : J {1,..., N}, J = N n} J q S n with the inner problem (optimal redistribution) min{d(p, (J, q)) : q S n } for fixed index set J. The outer problem is a N P hard combinatorial optimization problem while the inner problem may be reformulated as a linear program. Again a reformulation as linear mixed-integer program is possible. An explicit formula for D J := min q Sn d(p, (J, q)) is no longer available, but the inner problem may be rewritten.

22 For B B rect we define the system of critical index sets I(B) by and write I rect := {I(B) = {i {1,..., N} : ξ i B} : B B rect } P (B) Q(B) = i I(B) p i j I(B)\J Then, the rectangular discrepancy between P and Q is α Brect (P, Q) = max I I rect p i i I Using the reduced system of critical index sets j I\J I rect(j) := {I \ J : I I rect }, every I I rect(j) is associated with a family ϕ(i ) I rect : q j q j.. ϕ(i ) := {I I rect : I = I \ J} (I I rect(j)).

23 With the quantities γ I := max I ϕ(i ) i I p i and γ I := min p i I ϕ(i ) i J we obtain D J as infimum of the linear program min λt α + (1 λ)t ζ t α, t ζ 0, q j 0, j J q j = 1, η i,j 0, i = 1,..., N, j J, t ζ i=1,...,n,j J ĉr(ξ i, ξ j )η i,j, (I I rect(j)), j J η i,j = p i, i = 1,..., N, N i=1 η i,j = q j, j J, j I q j t α γ I, I I rect(j) j I q j t α + γ I, I I rect(j) We have I rect(j) 2 n and, hence, the LP should be solvable at least for moderate values of n.

24 How to determine I rect(j), γ I and γ I? Observation: I rect(j), γ I and γ I are determined by those rectangles B R, each of whose facets contains an element of {ξ j : j J}, such that it can not be enlarged without changing its interior s intersection with {ξ j : j J}. The rectangles in R are called supporting. Non supporting rectangle (left) and supporting rectangle (right). The dots represent the remaining scenarios ξ j, j J.

25 Proposition: It holds that I rect(j)= B R{I {1,..., N}\J : j I {ξ j }={ξ j :j J} int B} and, for every I Irect(J), γ I = max {P (int B) : B R, j I {ξ j } = {ξ j : j J} int B} γ I = p i, i I where I := {i {1,..., N} : min j I ξ j,l ξ i,l max j I ξ j,l, l = 1,..., d}. Note that R ( ) n+2 d 2!

26 Numerical results Comparison of scenario reduction for a Fortet-Mourier metric and a composite distance including α rect.

27 Optimal redistribution: α Brect versus ζ scenarios chosen by Quasi Monte Carlo out of 1000 samples from the uniform distribution on [0, 1] 2 and optimal probabilities adjusted w.r.t. λα Brect + (1 λ)ζ 2 for λ = 1 (gray balls) and λ = 0.9 (black circles)

28 Optimal redistribution w.r.t. the rectangular discrepancy α Brect : d n=5 n=10 n=15 n=20 N= N= Running times [sec] of the optimal redistribution algorithm The majority of the running time is spent for determining the supporting rectangles, while the time needed to solve the linear program is insignificant.

29 Optimal scenario reduction Forward selection: Step [0]: J [0] :=. Step [i]: l i argmin l J [i 1] inf q S i d λ J [i] := J [i 1] {l i }. Step [n+1]: Minimize d λ ( P, j J [n] q jδ ξj ) (P, ) q jδ j J[i 1] {l} ξj, s.t. q S n. N=100 n=5 n=10 n=15 d = d = d = Growth of running times (in seconds) of forward selection for λ = 1 Search for more efficient heuristics

30 Alternative heuristics (for P with independent marginals): (next neighbor) Quasi Monte Carlo: The first n numbers of randomized QMC sequences provide n equally weighted points. The closest scenarios are determined and the resulting discrepancy to the initial measure is computed for fixed probability weights. (next neighbor) adjusted Quasi Monte Carlo: The probabilities of the closest scenarios are adjusted by the optimal redistribution algorithm to obtain a minimal rectangular discrepancy to P. For general distributions P with densities transformation formulas are needed (e.g. Hlawka-Mück 71).

31 Conclusion: (Next neighbor) readjusted QMC decreases significantly the approximation error. Forward selection provides good results, but is very slow due to the optimal redistribution in each step. 0.8 discrepancy time Left: The distance d λ (λ = 1) between P and uniform (next neighbor) QMC points (dashed line) and (next neighbor) readjusted QMC points (solid line), and running time in seconds of optimal redistribution. Right: Distances α rect (solid) and ζ 2 (dashed) of 10 out of 100 scenarios, resulting from forward selection for several λ [0, 1].

32 Conclusions and outlook There exist reasonably fast heuristics for scenario reduction in linear twostage stochastic programs, It may be worth to study and compare exact solution methods with heuristics, Recursive application of the heuristics apply to generating scenario trees for multistage stochastic programs, For scenario tree reduction the heuristics have to be modified. For mixed-integer two-stage stochastic programs heuristics exist, but have to be based on different arguments. They are more expensive and so far restricted to moderate dimensions. This motivates to study exact approaches. There is hope for generating scenario trees for mixed-integer multistage models, but it is not yet supported by stability results.

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